Time series - Core Foundations

Time Series: Definition and Basic Concepts What Is a Time Series? A time series is simply a sequence of data points collected in chronological order. The key distinguishing feature of time series data is that observations follow a natural ordering based on when they were measured. For example, daily stock prices, monthly unemployment rates, or annual rainfall measurements are all time series. In most practical applications, time series observations are recorded at equally spaced time intervals. This regularity is important because it allows us to work with consistent, predictable gaps between measurements. Time series data can span remarkably different scales—from microseconds in high-frequency financial data to centuries in climate records. The image above shows a classic example: it displays measurements over time with visible irregular fluctuations around an underlying trend (shown in red). Why Time Series Are Different Time series data stand apart from other types of data in one crucial way: there is a natural temporal ordering that matters. This is fundamentally different from cross-sectional data, where each observation (say, the height of different students in a classroom) has no inherent ordering. You could rearrange cross-sectional observations randomly without losing any information, but rearranging a time series destroys everything important about it. Another key characteristic is autocorrelation: observations that occur close together in time tend to be more strongly related to each other than observations far apart. If today's stock price is $100, tomorrow's price is more likely to be near $100 than to be dramatically different. This autocorrelation structure is what makes time series analysis distinct from analyzing independent observations. In statistical terms, we often model a time series as a stochastic process—a random process that evolves over time. This framework helps us capture both the systematic patterns and the randomness inherent in real-world data. The Goals of Time Series Work Time series analysis serves two main purposes: Time series analysis seeks to understand what has happened in the data. This involves extracting meaningful statistics and uncovering underlying patterns. Three types of patterns commonly appear: Trends: long-term upward or downward movements Seasonal effects: regular patterns that repeat at fixed intervals (for example, retail sales that spike every December) Irregular fluctuations: random or unpredictable variations Time series forecasting takes this understanding further by building a model that predicts future values based on past observations. A forecasting model learns the patterns in historical data to estimate what will likely happen next. Stationarity: A Fundamental Concept Before we can effectively analyze or forecast a time series, we need to understand whether the data's statistical properties are stable or changing. This is where stationarity becomes critical. Strict and Wide-Sense Stationarity Strict stationarity means that the joint probability distribution of the time series does not change over time. More simply: if you take any chunk of data from your series and compare its statistical properties to any other chunk, they should be identical regardless of when each chunk occurred. This is a very strong requirement. Wide-sense (or second-order) stationarity is a weaker, more practical condition that's easier to check in real data. A time series is wide-sense stationary if: The mean is constant over time (doesn't drift up or down) The autocovariance depends only on the lag—the distance between observations—not on the specific time points For example, the autocovariance between measurements that are 5 days apart should be the same whether we're looking at measurements from January or July. Why does this matter? Many standard forecasting methods and statistical tests assume stationarity. If your data is non-stationary, these methods can give misleading results. Non-Stationary Series and Transformations Real-world time series frequently violate stationarity assumptions. A series might have a clear trend (the mean increases over time) or changing volatility (the variability increases or decreases). These are non-stationary properties. The good news is that non-stationary series can often be transformed into stationary ones. The most common approach is differencing: instead of analyzing the raw values, we analyze the changes between consecutive observations. This figure illustrates differencing: the top panel shows a non-stationary series with an obvious trend. The middle panel shows the yearly change (the difference between each year and the previous year), which removes the trend and creates a more stationary series. The bottom panel shows percentage change, another transformation option. Other transformations—like taking logarithms, square roots, or other mathematical functions—can also help stabilize a series before analysis. The choice of transformation depends on the specific characteristics of your data. <extrainfo> Seasonal and Seasonally Stationary Series Some time series exhibit regular, predictable patterns that repeat at fixed intervals. A series might be seasonally stationary, meaning it has consistent seasonal patterns but no overall trend. For example, ice cream sales consistently spike in summer and dip in winter, year after year. While seasonal patterns are important to recognize and model separately, the process of making a series stationary (through differencing or other methods) often handles seasonal non-stationarity along with trend-driven non-stationarity. </extrainfo> Ergodicity: Why a Single Series Can Be Enough Ergodicity is a technical property that has practical importance. An ergodic time series has a special property: time averages (calculated from a single long sequence of observations) converge to ensemble averages (calculated from many different possible realizations of the same process). What does this mean practically? In many statistical situations, we'd like to calculate properties of a distribution by taking many independent samples. With time series data, we usually have only one realization (one path through time) rather than many independent samples. Ergodicity tells us that's okay—we can reliably estimate statistical properties from that single long time series. This is why a long historical record of a time series can be statistically useful even though it's technically just one sequence of observations.